Having high hopes, finish off drug (7)

I believe the answer is:

aspirin

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'drug' is the definition.
(aspirin is a kind of drug)

I don't understand how the remainder of the clue works.

(Other definitions for aspirin that I've seen before include "Headache killer?" , "Pain-relief drug" , "Common drug in Paris" , "Analgesic over in Paris" , "Synthetic analgesic" .)

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